GIF Discontinuity Concept — JEE Maths in 1 Minute ⚡

❓ Concept

🎬 Discontinuity Trick for Greatest Integer Function | JEE in 60 Sec

GIF (floor function) dikha?
Samajh jao — continuity ka game start ho gaya hai 🔥
Is concept ko pakad liya, toh GIF ke 90% questions automatic clear.

---

🖼️ Concept Image

---

✍️ Short Explanation

Students ki sabse badi galti:
❌ “GIF non-integers par toot-ta hai”

Reality:
👉 GIF non-integers par continuous hota hai
👉 Sirf integers par jump aata hai

Bas inside expression ko dhyaan se dekho — answer wahin se niklega.

---

🔹 Step 1 — GIF Breaks Only at Integers (CORE RULE 💯)**

Greatest Integer Function $[x]$:

• Continuous at every NON-integer

• Discontinuous ONLY at INTEGERS

📌 Important correction:
Function ka argument integer hona chahiye,
na ki $x$ khud.

👉 So first rule:

Find where the INSIDE expression becomes an integer.

---

🔹 Step 2 — Break the Function into Parts

Typical JEE form:

$$f(x)=[\text{something}_1]-[\text{something}_2]$$

Method:

• Expression ko parts mein tod do

• Har GIF ko separately analyse karo

📌 Reason:
Har GIF apna independent jump create karta hai.

---

🔹 Step 3 — Check Both Expressions Separately

For each GIF:

• Solve:

$$\text{something}_i = \text{integer}$$

• Jitne x-values milte hain
  → utne potential discontinuity points

Then:

• Dono expressions ke x-values ka union lo

📌 Distinct points count karo
(not total equations)

---

🔹 Step 4 — Confirm the Jump (Safety Check 🧠)**

Har suspected x par:

• Left Hand Limit: $f(x^-)$
  

• Right Hand Limit: $f(x^+)$
  

If:

$$f(x^-)\neq f(x^+) \Rightarrow \text{DISCONTINUITY}$$

📌 GIF questions mein:

• Jump usually 1 unit ka hota hai

• Cancellation rare hoti hai (but possible!)

---

🔹 Step 5 — Domain Matters (EXAM TRAP ⚠️)**

Count ONLY those discontinuity points jo:

• Given interval ke andar ho

• Endpoints ko bhi check karo (open/closed interval!)

Example:

$$x\in[0,4]$$

Toh:

• Negative x ❌ ignore

• $x>4$ ❌ ignore

📌 Interval filter last mein lagana mat bhoolna.

---

⭐ Golden Summary Box (WRITE BOLD)

GIF Discontinuity Rule:
👉 $f(x)$ is discontinuous ONLY when the inside expression hits an INTEGER.

Bas:
1️⃣ Inside expression = integer solve karo
2️⃣ Distinct x-values nikalo
3️⃣ Interval ke andar count karo
4️⃣ (Optional) LHL ≠ RHL confirm karo

---

✅ Final Takeaway

🧠 GIF Master Rules

• GIF continuous at non-integers

• Discontinuous only at integer hits

• Composite GIF ⇒ analyse each part

• Union of all hit-points

• Domain filter at the end

Is logic ko pakad liya,
👉 Continuity & Differentiability ke GIF questions JEE mein free marks ban jaate hain 🚀

---